How to cut a cross shape into a sphere?
I would use Boolean Modifier. 3 times to be exact.

First cut a sphere shape out of a panel.
Do this by moving the panel into the sphere, until the edge of the panel is in the center of the sphere. Then add a Boolean Modifier, select the sphere as the Object and Apply.

(the panel might have missing faces, or an open side. If this happens you'll have to close up the mesh first I'm afraid.)

Then move the panel with the piece cut out of it into a Sphere, only a tiny bit deeper than the previous panel and put a Boolean Modifier on the Sphere this time (and select the panel as the Object), and again apply.

Then rotate the panel 90 degrees and do it again.

(To finish it off you could Shade Smooth the object and putt an Edge Split Modifier on it, to make it look nicer.)